Wind and Solar Park Yield

What is modelled

Wind speeds over a year follow a Weibull distribution. Each turbine converts wind power to electricity with a power curve: nothing below the cut-in speed, a cubic rise to the rated speed, flat output up to the cut-out speed. When the wind blows along the row, each rotor leaves a slower wake behind it that the next turbines sit in. The solar field adds energy from the peak sun hours of the day.

Equations

• Weibull density f(v) = (k/c)(v/c)k−1 e−(v/c)k, mean c Γ(1 + 1/k).
• Power P = ½ ρ Cp A v³ between 3 and 12 m/s, rated above 12 m/s, zero above 25 m/s; Betz Cp = 4a(1−a)² ≤ 16/27.
• Jensen wake deficit δ = (1 − √(1−Ct)) / (1 + 2 kw x / D)², several wakes added in quadrature.
• Yearly energy 8760 ∫ P(v (1 − δ)) f(v) dv; solar day (2/π) Ipeak L.

Checks run in the page

• Weibull mean and total probability against the Gamma function.
• A random-sample energy estimate against the integral.
• The Betz maximum by scanning the induction factor.
• The half-sine solar day against the peak-sun-hours formula.

Honest simplifications

The wind blows exactly along the row (the worst case), the power coefficient and thrust coefficient are fixed, the wake model is the simple top-hat Jensen model and the 3D scene is not copied one-to-one. Real parks have changing wind directions, turbulence, electrical losses and downtime, so these yields are estimates for teaching.

• Press Tight rows and watch the downstream turbines lose a large share of their energy.
• Press Windy ridge and see the capacity factor climb as the Weibull scale grows.
• Press Single turbine to see the undisturbed energy and no wake loss at all.